Observable Effects of Noncommutative Spacetime on the Hydrogen Atom

0.07 MB·English

by  Benjamin Johnson

OBSERVABLE EFFECTS OF NONCOMMUTATIVE SPACETIME
ON THE HYDROGEN ATOM
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0 Benjamin Katsuo Johnson1
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2 Department of Physics

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University of Maryland
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College Park, Maryland 20742

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3 University of Maryland Preprint: PP 06-001

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Abstract
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e We present a brief historical introduction to the motivations behind quan-
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tum mechanics and quantum field theory on noncommutative spacetime and
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s provide an insightful technique, readily accessible to the undergraduate stu-
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h dent, to examine the measurable effects of noncommutative spacetime on the
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familiarhydrogenatom. Finally, wecompareourresultstothosederivedfrom
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1 more sophisticated approaches.
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1 Introduction
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Noncommutativity is a fundamental concept underlying quantum physics. All con-
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c jugate variables fail to commute and indeed the uncertainty principle itself,
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y h¯
h (∆x)(∆p) ≥ (1)
p 2
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is based on the commutation relation of position and momentum,
i
X
r [x,p] ≡ xp−px = ih¯. (2)
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Quantum field theory on noncommutative spacetime is a modification of quan-
tum field theory based on the conjecture that spacetime coordinates at the Planck
scale do not commute. Traditionally, it is assumed that spacetime coordinates are
commutative,
[xµ,xν] = 0, (3)
1email address: [email protected].
whereas noncommutative spacetime posits that the commutator of the spacetime
coordinates does not vanish. Many authors assume
[xµ,xν] = iθµν, (4)
where θµν is a fixed antisymmetric numerical matrix; while others assume that all
thecommutatorsinvolving spacetime coordinatesandmomenta donotvanish. M.R.
Douglas and N.A. Nekrasov[1] and R.J. Szabo[2] give reviews of this subject.
W. Heisenberg first proposed the use of noncommuting spacetime coordinates
to reconcile singularity problems connected with ultraviolet divergences in the early
days of quantum electrodynamics. The hope was that smearing out the spacetime
coordinates would remove the ultraviolet divergences of quantum field theory associ-
ated with very short distances in spacetime. H. Snyder[3] formalized and published
the first paper on the subject, but it was largely overlooked both because it did
not solve the divergence problem and because of the the contemporaneous success
of renormalization techniques. Our approach stems from an argument proposed by
S. Doplicher, et al:[4] a high momentum probe must be employed to measure the
position of a particle with great precision. Since momentum and energy are depen-
dent on the same quantities, the process would create a system with enough energy
to bring about gravitational collapse, forming a black hole. This creates a minimal
observable length, deforming the traditional uncertainty relation by adding an extra
term,
h¯
(∆x)(∆p) ≥ +φ(∆p)2, (5)
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where h¯ is the reduced Planck’s constant, h/2π,
(λa )2
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φ = , (6)
h¯
where λ is a dimensionless parameter whose importance will be revealed in the
following section and a is the Bohr radius. We will use Eq. (5) to examine the
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effects of noncommutative spacetime on the hydrogen atom. More recently, string
theoretic considerations by N. Seiberg and E. Witten[5] revived interest the subject,
but their discussion is beyond the scope of this paper.
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2 The Traditional Hydrogen Ground State
We begin by following an argument from The Feynman Lectures on Physics[6] em-
ploying the uncertainty principle to approximate the size of the hydrogen atom.
From a classical perspective, the attractive electromagnetic force between the pro-
ton and electron in a hydrogen atom would cause the electron to radiate all of its
energy in the form of light and come crashing down into the nucleus. In quantum
theory, this cannot happen because it would violate the uncertainty principle, as we
would be able to accurately determine both the position and momentum of the elec-
tron. Thus we can assume the electron will orbit the nucleus, having an uncertainty
in position (∆x). From the uncertainty principle, we can see that the uncertainty
in the electron’s momentum is (∆p) ≈ h¯/(∆x). We can then find the energy of the
electron by summing its kinetic and potential energies,
(∆p)2 1 e2
E = − , (7)
2m 4πǫ (∆x)
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where m is the mass of the electron, π is the ratio of a circle’s circumference to its
diameter, ǫ is the permittivity of free space and and e is the fundamental charge,
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and use the uncertainty principle to replace (∆x),
(∆p)2 1 e2(∆p)
E = − . (8)
2m 4πǫ h¯
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We can set dE/d(∆p) = 0 to find the minimum energy and ground state radius of
the hydrogen atom. This gives us
(∆p) = me2/(4πǫ )h¯ (9)
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and
(∆x) = (4πǫ )h¯2/me2 ≡ a = 0.529˚A, (10)
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which is the Bohr radius. Using these to solve for the ground state energy, we find
1 2 m2e4
E = − (11)
(cid:18)4πǫ (cid:19) 2h¯2
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which computes to -13.6eV, the measured ionization energy of the hydrogen atom.
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3 The Perturbed Hydrogen Ground State
We can now use this approach with the deformed uncertainty relation to determine
the effect of noncommutative spacetime on the hydrogen atom. We begin with the
modified commutation relation given by S. Benczik, et al,[7]
[x,p] = ih¯(1+βp2), (12)
where β = φ/h¯ , which leads to the modified uncertainty relation
(∆x)(∆p) ≈ h¯ +φ(∆p)2. (13)
To find the minimum uncertainty we take
(∆x) ≈ (h¯ +φ(∆p)2)/(∆p) (14)
and substitute it into Eq. (7):
(∆p)2 1 e2(∆p)
E = − . (15)
2m 4πǫ h¯ +φ(∆p)2
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We can rearrange this equation algebraically to make it more manageable:
h¯(∆p)2 +φ(∆p)4 −2me2(∆p)/4πǫ
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E = . (16)
2mh¯ +2mφ(∆p)2
This makes it much easier to take dE/d(∆p) = 0 and solve for (∆p), as we can
divide out the denominator after taking the derivative using the quotient rule. Even
then, we are left with a quintic equation in (∆p),
1 1
0 = − 4m2h¯e2 +4mh¯2(∆p)+ 4m2e2φ(∆p)2 +8mh¯φ(∆p)3 +4mφ2(∆p)5.
4πǫ 4πǫ
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(17)
However, assuming φ is small, (an assumption that will be justified in the following
section) we can find a solution with a first order pertubation approximation. Solving
for the first order pertubation, we obtain
h¯
(∆p) = (1−3λ2) (18)
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and
(∆r) = a (1+4λ2). (19)
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Substituting these values into Eq. (7) gives us
1 e2 1 λ2e2
E = − + . (20)
4πǫ 2a 4πǫ a
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Here we see the importance of λ from Eq. (6). The terms of the equation’s right
hand side have the same variables, except that the second has λ which acts as a
dimensionless scaling variable. Since λ is squared, the second term is always positive
and thus non-commutative spacetime would increase the radius and ground state
energy of the hydrogen atom.
4 Agreement with Literature
We can compare our result to recent work by Benczik, et al and F. Brau,[8] who
solve the Schr¨odinger equation to calculate the perturbed hydrogen spectrum. Brau
gives an explicit formula for the perturbed hydrogen spectrum,
mα2c2 λ2a2m3α4c4(4n−3(l+ 1))
E = − + o 2 , (21)
nl 2n2 5h¯2 n4(l+ 1)
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where α is the fine structure constant, e2/(4πǫ )h¯c, c is the speed of light, n is
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the principal quantum number, and l is the angular momentum quantum number.
Substituting e2/(4πǫ )h¯c for the fine structure constant and ground state values
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n = 1 and l = 0, Eq. (21) simplifies to
1 e2 1 λ2e2
E = − + , (22)
4πǫ 2a 4πǫ a
o o o o
which is exactly the same as Eq. (20)! Moreover, Brau does an order of magnitude
calculation using the finite size of the electron to put an upper bound on λ ≤
2.83×10−7 validating our assumption that φ is small and pertubation theory could
be employed with accuracy. The most sophisticated mathematical tools in our
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approach were a partial derivative and first order pertubation approximation, both
of which can be learned by a typical undergraduate student, yet our results matched
very well those derived through much more rigorous means. We can see from our
calculations that noncommuting spacetime coordinates would increase the size of
the hydrogen atom very slightly and can expect that more accurate experimental
data will allow us to decrease the upper bound or observe a non-vanishing minimal
length.
Acknowledgements
We thank Professor O.W. Greenberg for his apt tutelage and infinite patience.
References
[1] M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001).
[2] R.J. Szabo, Phys. Rep. 378, 207 (2003).
[3] H. Snyder, Phys. Rev. 71, 38 (1947).
[4] S. Doplicher, K. Fredenhagen and J.E. Roberts, Commun. Math. Phys. 172,
187 (1995) and Phys. Lett. B 331, 33 (1994).
[5] N. Seiberg and E. Witten, J. High Energy Phys. 09, 032 (1999).
[6] R.P. Feynman, R.B. Leighton and M.L. Sands, The Feynman Lectures on
Physics (Addison-Wesley Publishing Co. Inc., Reading, MA 1965), 1st ed.,
Vol. III 2-6.
[7] S. Benczik et al. Phys. Rev. D66, 026003 (2002).
[8] F. Brau, J. Phys. A32, 7691 (1999).
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